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Questions tagged [levy-processes]

Question related to Lévy processes, i.e. stochastically continuous processes with independent, stationary increments.

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1answer
567 views

Proving properties for the Poisson-process.

Define a Poisson process as a Levy process where the increments have a Poisson distribution with parameter $\lambda$*"length of increment". I want to prove these properties: It has almost surely ...
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0answers
224 views

Holomorphic extensions of characteristic functions

Let $\chi(\xi) := \mathbb{E}e^{i \xi X}$, $\xi \in \mathbb{R}^d$, be the characteristic function of a random variable $X$. It is widely known that $\chi$ admits a holomorphic extension to the strip $\{...
6
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1answer
1k views

The Lévy-Khintchine formula and integrability conditions of a random measure

I am trying to see the connection between the Lévy-Khintchine and the integrability conditions of a Lévy measure. The literature seems to always connect both, but I cannot make sense of this relation ...
6
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1answer
202 views

Central Limit Theorem for Lévy Process

I am reading a book, which uses the Central Limit Theorem of Lévy Processes $X_t$ without mentioning the exact theorem. Due to the infinite divisible property I can write $X_t$ as a sum of $N$ iid ...
6
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1answer
314 views

Stochastic Integration with respect to Cauchy Process?

I'm interested in a one-dimensional stochastic process: $$dX_t = f(X_t)dt + g(X_t) dZ_t$$ where $Z_t$ is a Cauchy process and $f,g$ are nice polynomials (I'm looking at an ODE that gets perturbed by ...
6
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1answer
367 views

When is the compensated Poisson random measure a martingale ? (extensions to sets not bounded from 0)

Assume you have a Lévy process X. Let $N(t,A)$ be defined as the number of jumps in the interval $(0,t]$, such that the jumps size $\Delta X_s \in A$. It can be shown that if $0 \ne \bar{A}$, then $...
6
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0answers
172 views

Inequality for Lévy SDE

Let $X_{s}^{t,x}$ denote the solution at time $s$ of an Ito SDE whose coefficients are Lipschitz continuous with initial condition $X_t=x$. Let $t\leq s\leq T<\infty$. The inequality $$ \mathbb{E}\...
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2answers
1k views

Good book that contains stochastic integration, martingales and Lévy-processes?

Does anyone know about any good and easy interoductory books which contins information about martingales, sotchastic integration and Lévy-processes? I have tried reading: http://www.cambridge.org/us/...
5
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1answer
135 views

Largest jumps of a spectrally positive $\alpha$-stable process

Let $X(.)$ be a (strictly) $\alpha$-stable process (with $\alpha \in (1,2)$). Assume also that $X(.)$ is spectrally positive (its Lévy measure is concentrated in $[0,+\infty)$). I am looking for a ...
5
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1answer
1k views

Stochastic Integral with respect to Compensated Poisson Process

Proposition: Let $N_t$ be an $\mathcal{F}$-Poisson process and $M_t=N_t-\lambda t$ its compensated process. Then for any $\mathcal{F}$-predictable bounded process $H_t$, the stochastic integral $$(H\...
5
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1answer
169 views

A proof by René Schilling that a continuous Lévy process is integrable

In his treatise "An Introduction to Lévy and Feller Processes" (arXiv link), Prof. Dr. René Schilling gives a short and seemingly straightforward proof for the claim that a continuous Lévy process is ...
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365 views

Proof of strong Markov Property of double sided Levy Process

Let $(X_t)_{t\in\mathbb{R}}$ be a Levy Process, i.e. $X_0 = 0$ a.s., $X$ has independent and stationary increments, and almost all paths $t\mapsto X_t(\omega)$ are right continuous with left hand ...
5
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1answer
181 views

Law of large numbers for a Subordinator.

Let $\left( X_{t}\right) _{t\geq0}$ be a subordinator with the Laplace exponent given by $$ \Phi\left( \lambda\right) =d\lambda+\int_{0}^{\infty}\left( 1-e^{-\lambda x}\right) \nu\left( dx\...
4
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1answer
328 views

Left-continuity of a Lévy filtration

The natural filtration $(\mathcal{F}_t^X)_{t\geq 0}$ of a Lévy process $X$ is right-continuous, but what about left-continuity? A Lévy process is quasi left-continuous at time $t$ which says that \...
4
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1answer
244 views

Characteristic function under risk neutral measure

I am trying to derive a characteristic function (in Levy-Khintchine form) of a compound Poisson process $X_T$ under a risk neutral measure $\mathbb{Q}$, using the Esscher transfrom to change the ...
4
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1answer
436 views

Computing quadratic variation for stable Levy flights with $0<\alpha<2$?

The wiki page on semi-martingales states that Every Lévy process is a semimartingale. and that The quadratic variation exists for every semimartingale. Let $X_t$ be a stable Levy process with $...
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0answers
148 views

Potential measure of the product of (independent $\alpha$-stable) subordinators

For a nondecreasing Levy process $\mathbf{X}$ with values in $[0,\infty)$ (i.e. a subordinator) Jean Bertoin defines the potential measure of $\mathbf{X}$ in his book "Levy processes" as follows (p. ...
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How to Show RV Related to Poisson Random Measure is a.s. Finite?

I'm really new to this area of random measures, and I'm a bit confused on how to get started on this problem. Let $\mu$ be a measure on $\mathbb{R}$ with $\mu(\left\{0\right\}) = 0$ and $\int_\...
3
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2answers
982 views

Jumps of Lévy process

Let $X:=(X_t)_{t\geq0}$ be a Lévy process with triple $(b,A,\nu)$. Is there any known relation between the "distribution" of its jumps and the Lévy measure $\nu$? E.g. can we express something like $\...
3
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1answer
1k views

Implication of Lévy-Khintchine theorem/representation

I have trouble understanding the use/implication of the Lévy-Khintchine theorem. One possible way to state it is the following: The characteristic function $\varphi$ is infinitely divisible if and ...
3
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2answers
550 views

Blumenthal-Getoor index for IG and $\Gamma$

It is well known that that the gamma and inverse gaussian distributions lead to Levy processes. For a Levy process with Levy measure $\nu$ one can define (I'm new to this, this is the first ...
3
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1answer
50 views

Time Changing a Levy process to get rid of multiple

Suppose I have a Levy process $Z_t\triangleq A*X_t$, where $A$ is a $d\times d$-matrix and $X_t$ is Levy also. Then under what time-change $T(t)$ is $$ Z_{T(t)}=X_t? $$ For example, if $X_t$ is (d=1)...
3
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1answer
145 views

Intuition behind a set of a assumptions about Lévy processes.

I'm reading this paper in mathematical finance where they start with three assumptions, which I cannot really relate to anything I know so I hope you can help to give me some intuition about what they ...
3
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1answer
152 views

Does $\mathbb{E}(|X_t|) = O(t)$ hold for a Lévy process $(X_t)_{t \geq 0}$?

For a Levy process $(X_t)_{t\geq 0}$, we have $\mathbb{E}[X_t]=t\mathbb{E}[X_t^1]$ and $\text{Var}(X_t)=t\text{Var}(X_t^1)$. Does the same hold for the first absolute moment, i.e. does $\mathbb{E}[|...
3
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1answer
103 views

Monotone property of transition density of rotational $\alpha$-stable process

For a Brownian motion $B_t$ in $\mathbb R^d$, the transition density of $B_t$ is the normal distribution $$P_x[B_t\in dy]=(2\pi t)^{-d/2}e^{-\frac{|x-y|^2}{2t}}dy$$ and obviously the density is ...
3
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1answer
156 views

Lévy Process existence of the expectation of the supremum of the past process.

Given a Lévy Process $X_{t}$ in $\mathbb{R}^{d}$, with $X_{t}^{*}:=\sup_{s\in[0,t]}|X_{s}|$. I want to show, that for $t>0$ with $E[|X_{t}|]<\infty$ for $t>0$, then $E[X_{t}^{*}]<\...
3
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1answer
60 views

Derivations in paper “Brownian Distance Covariance” and their intuition

I am reading a paper and I am stuck at following points. I tried to analyze but do not understand where to start. The paper "Brownian distance covariance" contains following lemma Notations: $<t,...
3
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1answer
148 views

Probability that a Lévy-process is unbounded, zero-one law?.

For a Lévy-process, I need to prove that the probability that the trajectories are bounded on $[0,\infty)$ is either 0 or 1. Can you please help me? (The author says that this is a consequence of ...
3
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1answer
53 views

Minimal value of probability according to the difference of a Levy-process

Can we conclude for a Levy-Process, that for all $\epsilon>0$ it holds that $\min_{s\in [0,t]} \mathbb P\left(\left|X_t-X_s\right|\leq \epsilon\right)>0$? Stochastic continuity doesn't seem to ...
3
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1answer
329 views

Poisson Process independent Wiener Process using singular measures

I was reading some stochastic calculus of Jump processes and saw the following result: If $W_t$ is Brownian and $N_t$ is Poisson both adapted to the $W_t$'s natural filtration then these processes ...
3
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1answer
232 views

Two stochastic processes with the same distribution inducing different measures

I am currently reading Strook's $\textit{Probability Theory: An Analytic View}$, and I am confused by the following statement on page 156: "I take for $D(\mathbb{R}^N)$ the measurable structure given ...
3
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1answer
111 views

Size of the jumps in Itô-Lévy processes

I am trying to make sense of the Lévy Itô decomposition, in particular, of a note I have found regarding the size of the jumps. From the Lévy decomposition we know that any Lévy process is a ...
3
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2answers
94 views

How to compute the levy path integral with zero potential?

In quantum mechanics, if we have the quantum particle moving in the potential $V$ then the quantum-mechanical amplitude $K(x_b,t_b| x_a,t_a)$ can be written as $$K(x_b,t_b|x_a,t_a)=\int_{x_{t_a}=x_a,...
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0answers
147 views

Central Limit Theorem for a Lévy Process (mild assumptions)

I want to discuss the Central Limit Theorem (CLT) of a Lévy process under some assumptions. The answer of the previous post Central Limit Theorem for Lévy Process required a wide base of knowledge, ...
3
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0answers
148 views

What is the Lévy measure of the Student's $t$-distribution?

It is known since the 1970's that the Student's $t$-distribution is infinitely divisible. We can therefore apply the Lévy-Khintchine representation to it, and define the Lévy measure associated to a ...
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0answers
120 views

Bounded L2 increments for an Ornstein Uhlenbeck type process

Let $Z$ be an increasing Levy process (i.e. a subordinator). Let $\lambda>0$ and consider the Ornstein Uhlenbeck type SDE $$ d V_t = - \lambda V_t dt + d Z_{\lambda t } $$ where the integral can e....
3
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1answer
298 views

Intuitive explanation why finite activity Lévy processes does not have finite moments

I have a question about levy-processes. Let us denote the Lévy measure $\nu$ defined on $\mathbb{R}^d\setminus\{0\}$ and with $N$ the Poisson random measure such that $E[N(t,S)]=t\,\nu(A)$ for all $t&...
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135 views

Why do two points never 'arrive at once' in a Poisson point process

In the following, all the measure spaces are endowed with the Borel $\sigma$-algebra corresponding to their topology (we take the usual topology on $[0,\infty)$). Let $E$ be a Polish space and let $\...
3
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3answers
406 views

Jumps of independent Lévy processes

Suppose I have two independent Lévy processes $(X_t)_t$ and $(Y_t)_t$, both not continuous. Is anyone familiar and can refer me to a result (or a counterexample) which states that ${\displaystyle \...
2
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1answer
281 views

Fixed-time Jumps of a Lévy process

If one defines a Levy process as a stochastic process $(X_t)_{t\geq0}$ that has independent and stationary increments with (a.s.) cadlag paths (hence a def. withouth stochastic continuity). How can I ...
2
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2answers
301 views

Ornstein-Uhlenbeck Process simulation bug

I think I found a bug in a programm somebody posted but I can't fix it. It is about the simulation of an Ornstein-Uhlenbeck Process. The problem is from this [article][1] & and from wikipedia from ...
2
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1answer
361 views

Etemadi's inequality

In another post an inequality referred to as "Etemadi's Inequality" is mentioned twice - in the original post as well as in the answer. However, the contexts of usage are such as to raise the question ...
2
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1answer
196 views

Application of Lévy–Khinchine formula

How can we express the characteristic functions of Wiener and Poisson processes by using the Lévy–Khinchine formula? I don't know how to find the characteristic functions of particular Levy ...
2
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1answer
385 views

Stochastic Continuity of a Lévy Process

Given the following definition of a Lévy Process: A Lévy Process on $\mathbb{R}^d$ is a $D([0,\infty);\mathbb{R}^d)$-valued random variable $X$ for which $\mathbb{P}\left(X(0)=0\right)=1$ and $$\...
2
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1answer
66 views

The Levy measure of a multivariate alpha-stable Levy motion

I am having difficulties to understand the form of the Levy measure of the multivariate Levy-stable motion. Let me start by defining the one dimensional motion in order to clarify my question. The ...
2
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1answer
91 views

Exponential martingales and changes of measure

Suppose $X$ is a subordinator (an increasing Levy process) with Laplace exponent $\Phi$, i.e. $$ \exp(-\Phi(\lambda)) = E(\exp(-\lambda X_1)). $$ Let $\mathcal{F} = (\mathcal{F}_t)$ denote the ...
2
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1answer
97 views

Lévy-Khintchine formula and Taylor expansion

I have trouble finding out why this condition $\int_{\mathbb{R}\backslash\{0\}} \min(1, x^2 ) \nu(dx) < \infty$ in the Lévy-Khintchine formula is necessary. The Lévy-Khintchine formula is ...
2
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1answer
491 views

Subordination of a Levy process when the “subordinator” is not nondecreasing

If $X_t$ is an $\mathbb{R}$-valued Levy process and $Z_t$ is an $\mathbb{R}$-valued subordinator, we know that $X_{Z_t}$ is also Levy process. My question is, are there processes $Z_t$ which are not ...
2
votes
2answers
97 views

Levy measure of borel sets away from $0$

The following is of Philip Protter at page 26 of the book Stochastic integration and Differential equations that I have not been able to proved yet. Let $X$ a Levy process, and $\Lambda$ a borel set ...
2
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1answer
63 views

convergence towards infinity of jumping times of Levy processes

The following is a remark of Philip Protter at page 26 of the book Stochastic integration and Differential equation that I have not been able to proved yet. Let $\Lambda$ be a borel set in $\mathbb{...